All Questions
Tagged with spectral-graph-theory linear-algebra
85 questions
1
vote
0
answers
100
views
PageRank in directed graphs: equivalence of iterative and eigenvalue methods
Given a directed graph $ G $ with $ n $ nodes, we can represent this graph using an adjacency matrix $ A $. The stochastic matrix $ S $ can be derived from the adjacency matrix using the following ...
- 4,058
0
votes
0
answers
15
views
Change in two spectral deviations due to edge deletion in a signed graph
Prove (or disprove) the following. Let $\Sigma=(G,\sigma)$ be a given signed graph. If $\lambda_1\ge\lambda_2\ge\cdots\ge \lambda_n$ and $\mu_1\ge\mu_2\ge\cdots \ge \mu_n$ are the eigenvalues of the ...
- 61
4
votes
1
answer
103
views
When do the nonzero eigenvalues of a directed graph Laplacian have the same absolute value?
Question: Let $G$ be a strongly connected directed graph on $n$ vertices with Laplacian $L(G)$. Then $L(G)$ has one zero eigenvalue $\lambda_1=0$ and $n-1$ nonzero eigenvalues $\lambda_2,\ldots,\...
- 145
0
votes
1
answer
131
views
Function of eigenvalues of Laplacian matrix
Let $G$ be a simple $n$-vertex graph and let $\mu_n\geq\mu_{n-1}\geq\dots\geq\mu_1$ be the eigenvalues of its Laplacian matrix, how can I find a function $$f(\mu_1,\mu_2,\dots\mu_n) \text{ such that } ...
- 53
0
votes
0
answers
36
views
A class of directed graph, when their minimal polynomial of the adjacency matrix matches the characteristic polynomial
We consider an unweighted directed simple graph, $G$, with a Hamiltonian cycle.
Q. Assume that the adjacency matrix of $G$ is non-singular. Do the characteristic and minimal polynomials of the ...
- 4,058
2
votes
1
answer
94
views
Testing for equal characteristic polynomials using a single determinant calculation
Let $A_1,A_2$ be $n\times n$ symmetric matrices over $\{0,1\}$, and let $p_1, p_2$ be their respective characteristic polynomials over the rationals.
If $p_1 \ne p_2$, then there is some positive ...
- 37.7k
2
votes
0
answers
81
views
Given a low-rank symmetric positive semidefinite matrix and a basis of its nullspace, is there a fast way to get the nonzero eigenvalues?
I have a (possibly dense) $k\times k$ real matrix $L = AA^T + B^T B$, a type of combinatorial Laplacian (self-adjoint, symmetric, positive semidefinite) of rank $(k-n)$ and possibly repeated nonzero ...
- 21
1
vote
0
answers
54
views
Controlling quantity related to Laplacian pseudo-inverse of Erdős–Rényi graph
Consider an $n$-node undirected graph $G = (V, E)$ equipped with weights $W$. Let $L$ be the weighted graph Laplacian matrix, i.e. $L_{ij} = -W_{(i,j)}$ for $(i,j)\in E$ and $L_{ii} = \sum_{j:(i,j)\in ...
- 11
3
votes
1
answer
148
views
Spectrum of the adjacency matrix of certain directed graphs
For an undirected graph $G$, its adjacency matrix $A_G$ is symmetric, and by the (consequence of) spectral theorem, each of its Jordan blocks has size $1$. This is not true for a general directed ...
- 161
0
votes
0
answers
83
views
When is the sum of matrices (circulant + [super upper triangular]) not diagonalizable?
By the circulant matrix $C \in M_n(\mathbb{R})$, we mean that
$$ C = \left[\begin{array}{c|c|c|c} e_n & e_1 & \cdots & e_{n-1} \end{array}\right] $$
where $e_1,\dots,e_n$ are the standard ...
- 4,058
2
votes
1
answer
133
views
Is the sum of the circulant matrix with a super upper triangular matrix diagonalizable?
By the circulant matrix $C$ in $M_n(\mathbb{R})$, we mean that
$$C=[e_n|e_1|\cdots|e_{n-1}]$$ where $e_1,\cdots,e_n$ are the standard basis vectors in $\mathbb{R}^n$. It is well-known that
$$C=\...
- 4,058
1
vote
1
answer
198
views
Directed graph whose adjacency matrix admits only 0 as eigenvalue
Let $G$ be a directed graph and let $P_i$
be its vertices. Let $A$
be the corresponding adjacency matrix of $G$, i.e. $a_{i,j}=1$
if and only if there is a directed edge from $P_i$
to $P_j$, ($a_{i,...
- 4,058
6
votes
1
answer
515
views
Non-diagonalizability of the adjacency matrix of a directed graph
Let $G$ be a directed graph with no multiple edges or loops and let $P_i$ be its vertices. Let $A$ be the corresponding adjacency matrix of $G$, i.e. $a_{i,j}=1$ if and only if there is a directed ...
- 4,058
0
votes
0
answers
125
views
On the absolute difference of the Laplacian eigenvalues of an unbalanced signed graph and its underlying graph
Let $\Sigma=(G,\sigma)$ be an unbalanced signed graph with the underlying connected graph $G=(V,E)$ and $\sigma:E\rightarrow \{-1,1\}$, the signing function. Let the Laplacian eigenvalues of $\Sigma$ ...
- 61
4
votes
0
answers
148
views
Derivative of characteristic polynomial of a graph and derivative of characteristic polynomial of a vertex-deleted subgraph have a common root
Let $G$ be a simple graph and $G-i$ be one of its vertex-deleted subgraphs. Let $\phi(G,x)$ and $\phi(G-i,x)$ be the characteristic polynomials of $G$ and of $G-i$ respectively, with respect to their ...
- 327
2
votes
0
answers
351
views
Spectral norm bound for lower triangular matrix
Let $A$ be a $0/1$ square matrix which can be permuted to a non singular or a singular lower triangular matrix. Determinant is either $0$ or $1$. Can we provide tighter upper bounds on its spectral ...
- 13.9k
6
votes
1
answer
744
views
Relationship between spectral gaps of adjacency and Laplacian matrices of graphs
Let $G$ be an undirected simple graph on $n$ vertices, with self-loops allowed, and with arbitrary positive edge weights $w_{u,v}$ (which is $0$ if there is no edge between $u$ and $v$).
Let $A$ be ...
- 150
6
votes
1
answer
312
views
Determinant of walk matrix for a skew-symmetric matrix of even order
Let $S=(s_{ij})$ be a skew-symmetric integral matrix of order $n$. We only consider the case that $n$ is even. Let $e$ be the all-one vector in $\mathbb{R}^n$. Define the walk matrix $$W(S)=[e,Se,\...
- 437
7
votes
1
answer
1k
views
Strategies for bounding the spectral norm of a tensor?
Let $A$ be a symmetric $k$-tensor over a real or complex vector field $W$. We may define its spectral norm $|A|$ by
$$|A| = \sup_{v\in W} \frac{|\langle A,x^{\otimes k}\rangle|}{|x|_2^k}.$$
(...
- 20.2k
1
vote
0
answers
126
views
What are the meaning of left singular vectors of an incident matrix?
Consider a graph G and its incident matrix $B\in R^{m\times n}$. We can compute the SVD of $B$ as $B=U^{\top}\Sigma V$. Note that the Laplacian matrix $L_G=B^{\top}B$, so the right singular vectors ...
- 19
1
vote
0
answers
127
views
Delocalization of eigenvectors of graph Laplacians
Let $(V,E)$ be an undirected, connected graph with $n$ nodes. The graph Laplacian is defined as $L = D - A$, where $D$ is the degree matrix and $A$ is the adjacency matrix. Let $0 = \lambda_1 < \...
- 41
2
votes
0
answers
206
views
The rank of a Laplacian-type matrix
Suppose that $M$ is an integer, symmetric matrix of order $n>2$ with the positive integers $K_1,\dotsc,K_n$ on its main diagonal, and with all the off-diagonal elements equal to $0$ or $1$ so that ...
- 23k
5
votes
1
answer
678
views
Finding zero-one vectors in the row space of a matrix
Suppose that $M$ is a square matrix with all elements on its main diagonal equal to $1$, and every row containing exactly two off-diagonal elements equal to $-1$; all other elements are equal to $0$. ...
- 23k
2
votes
0
answers
79
views
Upper bound for smallest eigenvalue of infinite family of graphs
Let $\left\{G_{n}\right\}_{n=1}^{\infty}$ be a sequence of regular simple connected graphs with at least one edge such that $G_i$ is an induced sub-graph of $G_{i+1}$ and is not equal to $G_{i+1}$.
...
- 133
4
votes
1
answer
2k
views
Relation of row sums to largest eigenvalue
I know that the largest eigenvalue of a graph is bounded between the minimal and maximal row sum of the matrix. If I have a $0-1$ symetric matrix (an adjacency matrix) and I know $k$ of the rows have ...
- 133
0
votes
0
answers
41
views
Orthogonality condition of symmetric matrix pencil
Let $P(\lambda)=\lambda M−L\in \mathbb{R}^{n \times n}$ be a matrix pencil with symmetric nonsingular matrix $M$ and $L$ is a weighted Laplacian matrix of a connected graph. Clearly $(0,1_n)$ is an ...
- 21
9
votes
3
answers
3k
views
What happens to eigenvalues when edges are removed?
I am stuck at the following :
Let $G$ be a graph and $A$ is its adjacency matrix.
Let the eigenvalues of $A$ be $\lambda_1\le \lambda_2\leq \cdots \leq \lambda_n$.
If we remove some edges from the ...
- 444
2
votes
2
answers
129
views
Question about eigenvalues of connectivity matrices for graphs [closed]
I'm a computer science student working on a research project that deals with computational study of atomic clusters. I'm using a graph based representation of the clusters using a binary connectivity ...
0
votes
0
answers
67
views
Singular values and the chromatic number
What relation, if any, is there between the singular values of the adjacency matrix ( or possibly incidence matrix) of a simple graph and its chromatic number. Typically, do we have Hoffmann type, or ...
- 2,089
3
votes
0
answers
919
views
Principal eigenvector of non-negative symmetric block matrix is approximated by a linear combination of the principal eigenvectors of the blocks
Let $
M \in \mathbb{R}^{n \times n} =
\begin{bmatrix}
A & B \\
B^T & C
\end{bmatrix}
$ for some nonnegative $A \in \mathbb{R}^{k \times k}, B \in \mathbb{R}^{k \times n-k}, C \in \mathbb{R}^{n-...
- 123
3
votes
1
answer
271
views
Local-Global Principle in Graph Spectrum
The question is a bit vague, but any ideas/directions will be appreciated.
Let us fix an $n$-vertex $d$-regular graph $G=(V,E)$. As I understand it, the eigenvalues of the adjacency matrix $A$ of $G$ ...
- 949
7
votes
1
answer
641
views
Lower bound on the eigenvalues of the Laplacian
I am looking for a graph for which $2 d_{i} < \mu_{i}$, for some index $i$, where $\mu_{1} \leq \mu_{2} \leq \dots\leq \mu_{n}$ are the eigenvalues of the Laplacian matrix $L(G)$ and $d_{1} \leq d_{...
- 123
9
votes
3
answers
356
views
Spectrum of orthogonality graph (2)
The orthogonality graph, $\Omega(n)$, has vertex set the set of $\pm 1$ vectors of length $n$, with orthogonal vectors being adjacent.
I am only interested when $4|n$, since otherwise $\Omega(n)$ is ...
- 421
5
votes
0
answers
397
views
spectrum of orthogonality graphs
The orthogonality graph $\Omega(n)$ with $2^n$ vertices is the graph with vertex set $\{-1,+1\}^n$, with two vertices being adjacent if and only if they are orthogonal (as vectors in the standard ...
- 421
1
vote
0
answers
51
views
Relation between nullity of components to its parent graph
Let $G$ be an undirected graph and the corresponding adjacency matrix be $A$. Let $v$ be a cut-vertex of $G$. Let $G_1, G_2,\dots, G_k$ are the connected components of the induced graph $G-v$ ( the ...
- 640
4
votes
0
answers
141
views
algebraic connectivity of a tree
Suppose that $T$ is a tree with $n$ vertices and $L$ is the Laplacian matrix of $T$ and $0=\mu_1 \leq \mu_2 \leq \cdots \leq \mu_n$ are laplacian eigenvalues.
I think the multiplicity of $\mu_2$ can ...
- 221
4
votes
1
answer
256
views
An elementary inequality for graph Laplacians
Let $G$ be an arbitrary graph on $n$ vertices and $\mathcal L$ be its Laplacian.
I need to show that
\begin{equation}\tag{$*$}
\langle \mathcal Lx,\mathcal L(|x|^{p-2}x)\rangle_{\mathbb R^n}\ge 0\...
- 3,388
5
votes
1
answer
1k
views
The spectrum of the discrete Laplacian
Consider a connected (we define connected components by defining the set of vertices where every vertex has one neighbour) sublattice $V$ of the square lattice $V \subset\mathbb{Z}^2.$
On this we ...
- 53
1
vote
0
answers
422
views
Difference between largest two eigenvalues of a graph Laplacian
The difference between the smallest eigenvalue and the next-smallest of a graph Laplacian (equivalently, the difference between the largest and next-largest of the random walk Markov chain on the ...
- 1,068
2
votes
0
answers
699
views
Simple random walk on a discrete torus - the eigensystem, reference
My problem concerns finding a reference in which the formulae for the eigenvalues and the corresponding eigenvectors ($n$ linearly independent eigenvectors!) for the transition matrix of a simple ...
user83658
1
vote
0
answers
43
views
Is $L'L_\text{in}+L_\text{in}'L$ positive semi-definite?
Assume that $A$ is the adjacency matrix of a strongly connected directed graph, that is, $A$ is non-negative and irreducible. Let $$L_\text{in}=D_\text{in}-A',\;L=D_\text{in}-A'+D_\text{out}-A$$ where ...
- 11
10
votes
0
answers
225
views
Cospectral mate of rhombic dodecahedron
I am wondering if the following pair of cospectral graphs was previously known.
The rhombic dodecahedron graph looks like this (graph6 string: 'M?????rrAiTOd_YO?'):
As far as I know, it was previously ...
- 2,339
5
votes
1
answer
1k
views
Intuition on Kronecker Product of a Transition Matrix
Let $T$ be a $N\times N$ transition matrix for a markov chain with $N$ states. Thus $T_{ij}$ is the probability of transition from state $i$ to state $j$ (and thus rows summing to one). Now consider ...
- 1,421
1
vote
1
answer
271
views
Jordan blocks of directed graphs
Let $G$ be a (possibly weighted) directed graph with $n$ vertices and let $P$ be its transition matrix. That is, $P = D^{-1}A$ where $A$ is the graph's adjacency matrix and $D$ is a diagonal matrix ...
- 225
5
votes
2
answers
212
views
colored graph characteristic polynomial
This was asked previously on stackexchange and it was suggested to bring it here where more specialists could see it.
Given the adjacency matrix $\mathbf{A}$ for a simple connected graph, the ...
- 51
1
vote
0
answers
167
views
Largest eigenvalue of signed graph
Let us consider a graph where edges can have weight 1 or -1, such a graph is called signed graph. In a signed graph, a cycle is called balanced cycle when product of weights on its edges is positive ...
- 640
7
votes
4
answers
1k
views
Minimum negative eigenvalue of zero-one matrices
The following question must have been answered decades ago.
For $n$ fixed, what is the most negative eigenvalue among all trace zero zero-one matrices (that is, all entries are either zero or one, ...
- 4,679
4
votes
1
answer
464
views
Behaviour of eigenspaces of adjacency matrices after a single change to the graph
Say I know the eigenvalues and eigenvectors of an adjacency matrix of an unweighted graph. Can I say anything about the eigenvalues and eigenvectors of an adjacency matrix of a graph with one extra ...
- 41
4
votes
1
answer
2k
views
Size of connected components of a graph via its spectrum
I know that we can determine the number of connected components of a graph from the eigenvalues of its Laplacian matrix. My question is:
Is there a way to understand the size of each connected ...
- 5,407
1
vote
2
answers
421
views
Spectral radius of a non-negative matrix after moving and replicating an element
Let $A$ be a non-negative square matrix and its spectral radius (i.e., it's largest eigenvalue) be $\rho(A)$. I need to do the following operation to $A$ and compare the resulting spectral radii.
...
- 355